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    "# The Discrete-Time Fourier Transform\n",
    "\n",
    "*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Comunications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
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    "## Summary of Properties, Theorems and Transforms\n",
    "\n",
    "The [properties](properties.ipynb), [theorems](theorems.ipynb) and transforms of the discrete-time Fourier transform (DTFT) as derived in the previous sections are summarized in the following. The corresponding tables serve as a reference for the application of the DTFT in the theory of signals and systems. Please refer to the respective sections for details."
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    "### Definition\n",
    "\n",
    "The DTFT and its inverse are defined as\n",
    "\n",
    "\\begin{align}\n",
    "X(e^{j \\Omega}) &= \\sum_{k = -\\infty}^{\\infty} x[k] \\, e^{- j \\Omega k} \\\\\n",
    "x[k] &= \\frac{1}{2 \\pi} \\int_{-\\pi}^{\\pi} X(e^{j \\Omega}) \\, e^{j \\Omega k} \\; d \\Omega\n",
    "\\end{align}\n",
    "\n",
    "for $k \\in \\mathbb{Z}$ and $\\Omega \\in \\mathbb{R}$."
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    "### Properties and Theorems\n",
    "\n",
    "The properties and theorems of the DTFT are given as\n",
    "\n",
    "| &#65279; | $x[k]$ | $X(e^{j \\Omega}) = \\mathcal{F}_* \\{ x[k] \\}$ |\n",
    "|:---|:---:|:---:|\n",
    "| [Linearity](properties.ipynb#Linearity) | $A \\, x_1[k] + B \\, x_2[k]$ | $A \\, X_1(e^{j \\Omega}) + B \\, X_2(e^{j \\Omega})$ |\n",
    "| [Real-valued signal](properties.ipynb#Real-valued-signals) | $x^*[k]$ | $X^*(e^{-j \\Omega})$ |\n",
    "| [Convolution](theorems.ipynb#Convolution-Theorem) | $x[k] * h[k]$ | $X(e^{j \\Omega}) \\cdot H(e^{j \\Omega})$ |\n",
    "| [Shift](theorems.ipynb#Shift-Theorem) | $x[k - \\kappa]$ | $e^{-j \\Omega \\kappa} \\cdot X(e^{j \\Omega})$ |\n",
    "| [Multiplication](theorems.ipynb#Multiplication-Theorem) | $x[k] \\cdot h[k]$ | $\\frac{1}{2 \\pi} X(e^{j \\Omega}) \\circledast_{2 \\pi} H(e^{j \\Omega})$ |\n",
    "| [Modulation](theorems.ipynb#Modulation-Theorem) | $e^{j \\Omega_0 k} \\cdot x[k]$ | $X(e^{j (\\Omega- \\Omega_0)})$ |\n",
    "| [Parseval's Theorem](theorems.ipynb#Parseval's-Theorem) | $\\sum_{k = -\\infty}^{\\infty} \\lvert x[k] \\rvert^2$ | $\\frac{1}{2 \\pi} \\int_{-\\pi}^{\\pi} \\lvert X(e^{j \\Omega}) \\rvert^2 \\; d\\Omega$ |\n",
    "\n",
    "\n",
    "where $A, B \\in \\mathbb{C}$, $\\Omega_0 \\in \\mathbb{R}$ and $\\kappa \\in \\mathbb{Z}$."
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    "### Selected Transforms\n",
    "\n",
    "DTFTs which are frequently used are given as\n",
    "\n",
    "| $x[k]$ | $X(e^{j \\Omega}) = \\mathcal{F}_* \\{ x[k] \\}$ |\n",
    "|:---:|:---:|\n",
    "| $\\delta[k]$ | $1$ |\n",
    "| $1$ | ${\\bot \\!\\! \\bot \\!\\! \\bot}\\left( \\frac{\\Omega}{2 \\pi} \\right)$ |\n",
    "| $\\epsilon[k]$ | $\\frac{1}{1 - e^{-j \\Omega}} + \\frac{1}{2} {\\bot \\!\\! \\bot \\!\\! \\bot}\\left( \\frac{\\Omega}{2 \\pi} \\right)$ |\n",
    "| $\\text{rect}_N[k]$ | $e^{-j \\Omega \\frac{N-1}{2}} \\cdot \\frac{\\sin \\left(\\frac{N \\Omega}{2} \\right)}{\\sin \\left( \\frac{\\Omega}{2} \\right)}$ |\n",
    "| $e^{j \\Omega_0 k}$ | ${\\bot \\!\\! \\bot \\!\\! \\bot}\\left( \\frac{\\Omega-\\Omega_0}{2 \\pi} \\right)$ |\n",
    "| $\\sin(\\Omega_0 t)$ | $\\frac{1}{2} \\left[ {\\bot \\!\\! \\bot \\!\\! \\bot} \\left( \\frac{\\Omega + \\Omega_0}{2 \\pi} \\right) + {\\bot \\!\\! \\bot \\!\\! \\bot} \\left( \\frac{\\Omega - \\Omega_0}{2 \\pi} \\right)  \\right]$ |\n",
    "| $\\cos(\\Omega_0 t)$ | $\\frac{j}{2} \\left[ {\\bot \\!\\! \\bot \\!\\! \\bot} \\left( \\frac{\\Omega + \\Omega_0}{2 \\pi} \\right) - {\\bot \\!\\! \\bot \\!\\! \\bot} \\left( \\frac{\\Omega - \\Omega_0}{2 \\pi} \\right)  \\right]$ |\n",
    "| $a^k \\epsilon[k]$ | $\\frac{1}{1-a e^{-j \\Omega}}$ |\n",
    "\n",
    "\n",
    "where $\\Omega_0 \\in \\mathbb{R}$ and $|a| < 1$. More transforms may be found in the literature or [online](https://en.wikipedia.org/wiki/Discrete-time_Fourier_transform#Table_of_discrete-time_Fourier_transforms)."
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    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples*."
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